×

A remark on estimates for uniformly elliptic operators on weighted \(L^p\) spaces and Morrey spaces. (English) Zbl 0939.35036

The authors consider a class of uniformly elliptic operators \(L=-\sum^n_{i,j=1} \partial_i(a_{ij} (x)\partial_j) +V(x)\), where \(V\) is a nonnegative function on \(\mathbb{R}^n\). Under certain assumptions, several estimates for \(VL^{-1}\), \(V^{1/2}\nabla L^{-1}\) and \(\nabla^2L^{-1}\) on \(L^p\) spaces and Morrey spaces are obtained.
Reviewer: J.Mo (Anhui)

MSC:

35B45 A priori estimates in context of PDEs
35J15 Second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Avellaneda, Comm. Pure Appl. Math. 44 pp 897– (1991)
[2] , and : Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdam, 1978
[3] Chiarenza, Rend. Mat. 7 pp 273– (1987)
[4] Coifman, Studia Math. 51 pp 241– (1974)
[5] Chiarenza, Proc. A. M. S. 98 pp 415– (1986)
[6] Fefferman, Bull. Amer. Math. Soc. 9 pp 129– (1983)
[7] Gehring, Acta. Math. 130 pp 265– (1973)
[8] Garcia-Cuerva, North-Holland Math. Studies 116 (1985)
[9] Gruter, Manus. Math. 37 pp 303– (1982)
[10] Hedberg, Proc. A. M. S. 36 pp 505– (1972)
[11] and : Une Inégalité L2 , preprint
[12] Karp, J. D’Analyse pp 275– (1995)
[13] Kurata, Indiana Univ. Math. J. 43 pp 411– (1994)
[14] and : Fundamental Solution, Eigenvalue Asymptotics and Eigenfunctions of Degenerate Elliptic Operators with Positive Potentials, to appear in Studia Math. · Zbl 0956.35058
[15] Mizuhara, ICM-90 Satellite, pp 183– (1991) · doi:10.1007/978-4-431-68168-7_16
[16] Murata, Pub. Res. Instit. Math. Sci. 26 pp 585– (1990)
[17] Shen, Indiana Univ. Math. J. 43 pp 143– (1994)
[18] Shen, Grenoble 45 pp 513– (1995) · Zbl 0818.35021 · doi:10.5802/aif.1463
[19] Smith, Duke Math. J. 63 pp 343– (1991)
[20] Tachizawa, Tohoku Math. J. 42 pp 381– (1990)
[21] Thangavelu, Comm. in P. D. E. (8) 15 pp 1199– (1990)
[22] : Harmonic Analysis for Some Schrödinger Type Operators, Ph. D. Thesis, Princeton University, 1993
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.